For a zero-temperature Landau symmetry-breaking transition in $n$-dimensional space that completely breaks a finite symmetry $G$, the critical point at the transition has the symmetry $G$. In this paper, we show that the critical point also has a dual symmetry—a $(n-1)$-symmetry described by a higher group when $G$ is Abelian or an algebraic $(n-1)$-symmetry beyond a higher group when $G$ is non-Abelian. In fact, any $G$-symmetric system can be viewed as a boundary of $G$-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk $G$-gauge theory gives rise to the symmetry and the dual symmetry, respectively. So any $G$-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry and the dual symmetry. However, part (and only part) of the categorical symmetry must be spontaneously broken in any gapped phase of the system, but there exists a gapless state where the categorical symmetry is not spontaneously broken. Such a gapless state corresponds to the usual critical point of Landau symmetry-breaking transition. The above results remain valid even if we expand the notion of symmetry to include higher symmetries and algebraic higher symmetries. Thus our result also applies to critical points for transitions between topological phases of matter. In particular, we show that there can be several critical points for the transition from the 3 + 1-dimensional $Z_2$ gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a $Z_2$ 0-symmetry and its dual, a $Z_2$ 2-symmetry, while the critical point of the confinement transition has a categorical symmetry formed by a $Z_2$ 1-symmetry and its dual, another $Z_2$ 1-symmetry.

• Received 31 December 2019
• Revised 3 June 2020
• Accepted 3 June 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033417

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Published by the American Physical Society